Haar measure for totally positive split real semisimple quantum groups of classical type

Abstract

Let $\hbar$ be a non-zero real number and put $q= e^{-πi \hbar}$. Let $G$ be a semisimple split real Lie group of classical type, with Lie algebra $\mathfrak{g}$ and associated Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$. We show that there exists a locally compact quantum group $(L_{\hbar}(B),Δ)$ which integrates the Borel part $U_q(\mathfrak{b})$ of the Drinfeld--Jimbo quantized enveloping algebra $U_q(\mathfrak{g})$, viewed as a Hopf $*$-algebra. We further compute the associated Haar weights and modular data for $(L_{\hbar}(B),Δ)$. Up to an excess Cartan part, the Drinfeld double of $(L_{\hbar}(B),Δ)$ can be seen as an analytic quantization of the function algebra of the totally positive semigroup $G^+\subseteq G$.